(a)If{sn}is increasing and unbounded, thensn→+∞.(b)If{sn}is decreasing and unbounded, thensn→ ∞.
Proof:
Cauchy Sequences
Definition 5.
A sequence
{
s
n
}
is a
Cauchy sequence
if to each
>
0
there is a positive integer
N
such that
m, n > N
implies

s
n

s
m

<
.
THEOREM 10.
Every convergent sequence is a Cauchy sequence.
Proof:
Suppose
s
n
→
s
. Let
>
0. There exists a positive integer
N
such that

s

s
n

<
/
2
for all
n > N
. Let
n, m > N
. Then

s
m

s
n

=

s
m

s
+
s

s
n
 ≤ 
s
m

s

+

s

s
n

<
2
+
2
=
.
Therefore
{
s
n
}
is a Cauchy sequence.
THEOREM 11.
Every Cauchy sequence is bounded.
19
Proof:
Let
{
s
n
}
be a Cauchy sequence. There exists a positive integer
N
such that

s
n

s
m

<
1
whenever
n, m > M
. Therefore

s
n

=

s
n

s
N
+1
+
s
N
+1
 ≤ 
s
n

s
N
+1

+

s
N
+1

<
1 +

s
N
+1

for all
n > N.
Now let
M
= max
{
s
1

,

s
2

, . . .,

s
N

,
1 +

s
N
+1
}
. Then

s
n
 ≤
M
for all
n
.
THEOREM 12.A sequence{sn}is convergent if and only if it is a Cauchy sequence.
Exercises 2.3
1. True – False.Justify your answer by citing a theorem, giving a proof, or giving a counterexample.(a) If a monotone sequence is bounded, then it is convergent.(b) If a bounded sequence is monotone, then it is convergent.(c) If a convergent sequence is monotone, then it is bounded.(d) If a convergent sequence is bounded, then it is monotone.2. Give an example of a sequence having the given properties.(a) Cauchy, but not monotone.(b) Monotone, but not Cauchy.(c) Bounded, but not Cauchy.3. Show that the sequence{sn}defined bys1= 1andsn+1=14(sn+ 5)is monotone andbounded. Find the limit.4. Show that the sequence{sn}defined bys1= 2andsn+1=√2sn+ 1is monotone andbounded. Find the limit.5. Show that the sequence{sn}defined bys1= 1andsn+1=√sn+ 6is monotone andbounded. Find the limit.6. Prove that a bounded decreasing sequence converges to its greatest lower bound.7. Prove Theorem 9 (b).II.4.SUBSEQUENCESDefinition 6.Given a sequence{sn}.Let{nk}be a sequence of positive integers such thatn1< n2< n3<· · ·. The sequence{snk}is called asubsequenceof{sn}.ExamplesTHEOREM 13.If{sn}converges tos,then every subsequence{snkof{sn}also convergestos.
20
CorollaryIf{sn}has a subsequence{tn}that converges toαand a subsequence{un}thatconverges toβwithα=β,then{sn}does not converge.
Every bounded sequence has a convergent subsequence.
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 Fall '15
 Almis
 Math, Real Numbers, Integers, Limit of a sequence, Sn